Kinetic energy contribution to the exchange-correlation energy functional of the extended constrained-search theory

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1 PHYSICAL REVIEW A 79, Kinetic energy contribution to the exchange-correlation energy functional of the extended constrained-search theory Katsuhiko Higuchi Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima , Japan Masahiko Higuchi Department of Physics, Faculty of Science, Shinshu University, Matsumoto , Japan Received 30 May 2008; revised manuscript received 5 August 2008; published 13 February 2009 We present the kinetic energy contribution to the exchange-correlation energy functional of the extended constrained-search ECS theory by means of the generalized Bauer s relation. Due to the nature of the exchange-correlation energy functional being a function of the Bohr radius and e 2, three kinds of expressions for the kinetic energy contribution are obtained. These can be utilized as constraints in developing and/or evaluating the approximate form of the exchange-correlation energy functional of the ECS theory. Furthermore, by combining three expressions with the virial relation, we derive other useful relations that include not the kinetic energy contribution but only the exchange-correlation energy functional. DOI: /PhysRevA PACS numbers: Mb I. INTRODUCTION In the framework of the density-functional theory DFT 1,2, the constrained-search formulation 3 5 was originally proposed by Levy, and was extended to specific cases by several workers. For instance, the electron density and off-diagonal elements of the first-order reduced density matrix were treated as basic variables in Levy s original paper 3. Also, the extensions of the constrained search formula to the spin-density-functional theory SDFT 6, the current-density-functional theory CDFT 7, the pair density-functional theory PDFT 8,9, and the symmetryadopted version 10,11 have been done so far. Thus, the constrained-search formulation has been pursued in specific cases. We have previously proposed the generalized framework of the constrained-search formulation, in which arbitrary physical quantities can be chosen as basic variables in addition to the electron density In this framework, the extended version of the constrained-search formula is utilized in defining the universal functional and the kinetic energy functional. Therefore, we call the framework the extended constrained-search ECS theory. The validity of the ECS theory has been confirmed by revisiting various kinds of previous theories 12,13 such as the SDFT 15,16, CDFT 17 24, the Hartree-Fock-Kohn-Sham scheme 25,26, and the LDA+U method The ECS theory has the possibility to provide a specially made theory for each individual system, because quantities that characterize the ground state of the system can be chosen as basic variables. Indeed, various kinds of quantities, for example the pair density 31 36, the local density of state at the Fermi level 13, and the spin current density 37 have been chosen as basic variables so far. In order to perform electronic structure calculations by means of the ECS theory, we have to develop the approximate form of the exchange-correlation energy functional. For this aim, the discussion on how to develop the exchangecorrelation energy functional of the conventional DFT is a good reference. There are, in general, two strategies for developing the approximate form of the exchange-correlation energy functional 14,38. One is to utilize the couplingconstant integration expression for the exchange-correlation energy functional. Various kinds of approximations have been proposed on the basis of the coupling-constant expression in previous theories 2,17,18, These include the local-density approximation 2,17,18,38, the weighteddensity approximation 38,40 44, and the average-density approximation Along this strategy, we have already derived the coupling-constant integration expression in the ECS theory 14. This expression is one of the good starting points toward the development of the approximate form. Another strategy is to employ as constraints exact relations that are fulfilled by the exchange-correlation energy functional. In accordance with this strategy, approximate forms of the exchange and correlation energy functionals have been developed in the previous theories. For example, the generalized gradient approximation GGA 45 47, the density-moment expansion method 48 52, the local and variable-separation approximation 53, and the vorticity expansion approximation VEA have been developed in the DFT, SDFT, and/or CDFT. Concerning the exchangecorrelation energy functional of the ECS theory, some exact relations have already been obtained 14,54. We have presented the virial relation in the ECS theory 14. Furthermore, by using the coordinate scaling of electrons and adiabatic connection, Nagy has derived exact relations and equations of hierarchy in the ECS theory 54, similarly to the case of the DFT 55,56. Along the latter strategy, exact relations that are associated with the kinetic energy contribution to the exchangecorrelation energy functional have also been derived in the conventional DFT 26, Since it is on the same order of magnitude as the correlation energy for atomic and molecular systems 26,57, the kinetic energy contribution is one of the important issues in the conventional DFT. Bass 58 has derived its useful formula within the conventional DFT by using Bauer s relation 59. Also, Görling, Levy, and /2009/792/ The American Physical Society

2 KATSUHIKO HIGUCHI AND MASAHIKO HIGUCHI Perdew have revisited both Bauer s relation and Bass s expression within the constrained-search formulation of the DFT 61. An alternative expression for this kinetic energy contribution can be obtained from the virial theorem 60. In this paper, the kinetic energy contribution to the exchange-correlation energy functional is derived in the framework of the ECS theory by generalizing Bauer s relation. Furthermore, we shall derive the set of exact relations that will be useful in developing and/or evaluating the approximate form of the exchange-correlation energy functional of the ECS theory. The organization of this paper is as follows. In Sec. II, a generalized version of Bauer s relation is derived in the framework of the ECS theory. In Secs. III and IV, the relation is utilized to present the kinetic energy contribution to the exchange-correlation energy functional of the ECS theory. Three kinds of expressions are presented. Furthermore, some exact relations that can be used as constraints to the exchange-correlation energy functional of the ECS theory are derived in combination with the virial relation. Finally, in Sec. V, we summarize the results and give some comments on them. II. GENERALIZATION OF BAUER S RELATION TO THE ECS THEORY With reference to the derivation procedure of Bauer s relation 59,61, let us start with the Hamiltonian that is augmented by an arbitrary operator Ô via a scalar field, Ĥ = Tˆ + gŵ + v ext rˆrdr + Ô, where Tˆ, Ŵ, and ˆr are operators of the kinetic energy, electron-electron interaction, and electron density, respectively, and where g denotes the coupling constant for the electron-electron interaction. It should be mentioned that if and g are equal to 0 and 1, respectively, then Eq. 1 becomes the actual Hamiltonian of a many-electron system in an external potential v ext r. In the ECS theory 12 14, arbitrary physical quantities can be chosen as basic variables in addition to the electron density r. Here, suppose that the basic variable chosen is denoted by Xr. Using the ECS formula, the universal functional is defined as F,X,g = Min,X Tˆ + gŵ + Ô, where,x indicates that the minimization is performed among all antisymmetric wave functions which yield the prescribed r and Xr. Of course, Eq. 2 becomes the usual universal functional F,X in the case of,g =0,1, i.e., F,X0,1=F,X. It should be noted that since the necessary and sufficient conditions for N-representability of (r,xr) are unknown except for in specific cases, there may not exist some that is associated with (r,xr) of interest. One possible way to avoid this difficulty was recently proposed by Levy and Ayers 62. Here we shall give a comment on the existence and uniqueness of the minimum on the right-hand side of Eq It is shown that the minimum exists for v-representable (r,xr), but it is an open question whether the minimum exists or not for (r,xr) that is both non-v-representable and N-representable. Also, whether the minimum is unique or not is the remaining issue that should be discussed. Although such problems are pointed out and discussed in specific cases 4,5,10,11,15,63 72, we shall proceed to a discussion under the assumption that the minimum uniquely exists on the right-hand side of Eq. 2, i.e., (r,xr) is supposed to be of that type. If we assume the differentiability of Eq. 2 and denote the minimizing wave function in Eq. 2 as,x g, then,x g obeys the following equation: Tˆ + gŵ + Ô + v g rˆrdr + a g r Xˆ rdr,x g = E g,x g, where E g, v g r, and a g r are the Lagrange multipliers that correspond to constraints on,x g, i.e.,,x g is normalized to unity, and yields both r and Xr. From Eq. 3, E g is given by E g =,X gtˆ + gŵ + Ô + v g rˆrdr + a g r Xˆ rdr,x g. Differentiating both sides of Eq. 4 with respect to or g, we get E g 3 4 =,X gô,x g + v gr rdr + a gr Xrdr, 5 E g g =,X gŵ,x g + v gr rdr g + a gr Xrdr. 6 g Here, the Hellmann-Feynman theorem has been used. On the other hand, using Eq. 2, we can rewrite Eq. 4 as E g = F,X,g + v g rrdr + a g r Xrdr. Differentiating both sides of Eq. 7 with respect to or g, we also get E g = F,X,g PHYSICAL REVIEW A 79, a gr + v gr rdr 7 Xrdr,

3 KINETIC ENERGY CONTRIBUTION TO THE EXCHANGE- E g g = F,X,g g + a gr g + v gr rdr g Xrdr. 9 Comparison of Eq. 5 with Eq. 8, and that of Eq. 6 with Eq. 9, leads to F,X,g =,X g Ô,X g, 10 F,X,g =,X g Ŵ,X g, 11 g respectively. By utilizing Eq. 10, we obtain F,X,1 F,X,0 =0 =,X 01 Ô,X 01,X 00 Ô,X Here, we shall suppose that r and Xr are noninteracting v-representable, and that Xr is a single-particle property such as the spin density or paramagnetic current density. Under these assumptions,,x 00 is equal to the minimizing Slater determinant in the definition of the kinetic energy functional T s,x=min,x Tˆ, where denotes the Slater determinant. This is because these assumptions lead to the equation Min,X Tˆ =Min,X Tˆ, i.e., F,X0,0=T s,x 14. Ifwe denote the minimizing Slater determinant as,x, then we get,x 00 =,X. Under this assumption, Eq. 12 is rewritten as F,X,1 F,X,0 =0 =,XÔ,X,XÔ,X, 13 where we have used the fact that,x 01 is identical with the minimizing wave function,x in the definition of F,X =Min,X Tˆ +Ŵ. The left-hand side of Eq. 13 can also be obtained by integrating the left-hand side of Eq. 11 with respect to g from 0 to 1 and by differentiating the result with respect to. We have F,X,1 F,X,0 =0 = 1,X g Ŵ,X g dg= Substitution of Eq. 14 into Eq. 13 leads to,xô,x,xô,x = 1,X g Ŵ,X g dg= Next, we shall associate the right-hand side of Eq. 15 with the exchange-correlation energy functional of the ECS theory. For this purpose, we define the functional E xc,x by with E xc,x = F,X,1 F,X,0 U, 16 U = e2 rr drdr r r As mentioned above, since F,X0,1 and F,X0,0 are equal to F,X and T s,x, respectively, E xc,x becomes the exchange-correlation energy functional E xc,x of the ECS theory in the case of =0. Namely, we have E xc,x0 = E xc,x. Using Eq. 11, Eq. 16 is rewritten as 1 df,x,g E xc,x dg U =0 dg =0 PHYSICAL REVIEW A 79, ,X g Ŵ,X g dg U Since U is independent of, substitution of Eq. 19 into Eq. 15 leads to,xô,x,xô,x = E xc,x. 20 =0 If the ground-state values of basic variables are denoted by 0 r and X 0 r, then the ECS theory ensures that 0,X 0 is the ground-state wave function 0. In addition, 0,X 0 is the Slater determinant that is constructed from the solutions for the single-particle equation of the ECS theory. When r= 0 r and Xr=X 0 r, Eq. 20 becomes 0 Ô 0 0,X 0 Ô 0,X 0 = E xc 0,X =0 This is the generalization of Bauer s relation. The original Bauer s relation 59 has been successfully used for the derivation of the kinetic energy contribution to the exchange and correlation energy functional of the conventional DFT 58. In the following section, we shall utilize Eqs. 20 and 21 in order to derive the kinetic energy contribution to the exchange and correlation energy functional of the ECS theory. III. KINETIC ENERGY CONTRIBUTION TO E xc [,X] In the ECS theory, the kinetic energy contribution to E xc,x is expressed as T,X T s,x 12 14, where

4 KATSUHIKO HIGUCHI AND MASAHIKO HIGUCHI T,X and T s,x are given by,xtˆ,x and,xtˆ,x, respectively. In order to derive the kinetic energy contribution, let us choose Tˆ as Ô in Eq. 20. We have T,X T s,x = E xc,x. 22 =0 This is an expression for the kinetic energy contribution to E xc,x 73. It should be noticed that the right-hand side of Eq. 22 is an incalculable expression. In the DFT, the right-hand side of Eq. 22 is rewritten in a calculable expression by Bass 58. Namely, it is rewritten in terms of the Bohr radius a B dependence by treating a B as a variable parameter 58. The calculable expression by Bass is called a useful formula 74, and is applied to the homogeneous electron gas 58,74 and inhomogeneous systems 58,61 in evaluating the kinetic energy contribution to the exchangecorrelation energy functional. Therefore, it is undoubtedly meaningful to replace the incalculable expression Eq. 22 with a calculable one. In order to rewrite the right-hand side of Eq. 22 by a calculable expression, we shall formally treat e 2, m, and 2 as variable parameters, although they are fundamental physical constants. As shown below, this enables us to rewrite the right-hand side of Eq. 22 in terms of the e 2, m, or 2 dependence. In order to get the, e 2, m, and 2 dependences of E xc,x, it is necessary to know those of,x g. Since,X g is the minimizing wave function defined in Min,X 1+Tˆ +gŵ, the equation Min,X 1+Tˆ + gŵ =,X g 1+Tˆ + gŵ,x g holds. Dividing both sides of this equation by e 2, we notice that,x g is also the minimizing wave function in Min,X 1+ 2 /me 2 T ˆ +gw ˆ, where T ˆ and W ˆ are defined as T ˆ =m/ 2 Tˆ and W ˆ =1/e 2 Ŵ, respectively. Applying the Lagrange multiplier method, we get the equation for,x g, 1+ 2 me 2Tˆ + gŵ + vrˆrdr + ar Xˆ rdr,x g =,X g, 23 where, vr, and ar stand for the Lagrange multipliers. In order to discuss the, e 2, m, and 2 dependences of,x g, we assume that Xˆ r is normalized by a factor fe 2,m, 2 so that it is independent of e 2, m, and 2, i.e., Xˆ r is given by xˆr/ fe 2,m, 2 if xˆr is the usual physical operator. For example, instead of the paramagnetic current density ĵ p r, the normalized quantity m/ĵ p r is supposed to be chosen as Xˆ r. In Sec. IV, we will discuss this assumption again from a practical point of view. Under the above-mentioned assumption, we shall first consider the, e 2, m, and 2 dependences of the Lagrange multipliers vr and ar. Since these Lagrange multipliers are determined by requiring that,x g yields the prescribed basic variables, they can be denoted as v,xr and a,xr, respectively. For the present purpose, let us suppose that the Lagrange multipliers are determined for the prescribed r and Xr, and consider the case in which, e 2, m, and 2 are changed such that 1+ 2 me 2 is left invariant. In this case, the first and second terms of Eq. 23 are unchanged. If we tentatively substitute v,xr and a,xr into Eq. 23, then the left-hand side of Eq. 23 is exactly the same as before due to the assumption on Xˆ r. Then, we again obtain,x g as the solution. Since,X g yields the prescribed r and Xr, v,xr and a,xr are just Lagrange multipliers that are needed in this case. Thus, both v,xr and a,xr are left invariant under the changes such that 1+ 2 me 2 is left invariant. Namely, we can say that both v,xr and a,xr depend on, e 2, m, and 2 only through the factor 1+ 2 me 2. This leads to the important fact that,x g depends on, e 2, m, and 2 only through the factor 1+ 2 me 2. In order to express this dependency explicitly, we shall rewrite,x g as,x(1+ 2 /me 2,g). By using this expression, Eq. 19 is rewritten as E xc,x e 2 =0 PHYSICAL REVIEW A 79, ,g,X1+ 2 me Ŵ,X1+ 2 me 2,g dg 1 2 rr drdr. r r 24 The right-hand side of Eq. 24 can be regarded as a function of 1+ 2 me 2.IfẼ xc,x(1+ 2 /me 2 ) is defined as the right-hand side of Eq. 24, then we get E xc,x = e 2 Ẽ xc,x1+ 2 me Since E xc,x0 is equal to E xc,x as shown in Eq. 18, substitution of =0 into Eq. 25 leads to E xc,x = e 2 Ẽ xc,x 2 me This relation means that E xc,x is a function of the Bohr radius a B = 2 me 2 and e 2. Differentiating Eq. 25 with respect to, and changing e 2, m, and 2 as variables, we obtain three kinds of expressions for E xc,x/ =0, 2 E xc,x = E xc,x e E xc,x =0 e 2 2,m 27a = m E xc,x m 2,e 2 27b

5 KINETIC ENERGY CONTRIBUTION TO THE EXCHANGE- 2 = E xc,x 2. 27c e 2,m The reason why three different expressions are obtained is due to the fact that E xc,x is a function of a B and e 2 see Eq. 26. By substituting Eqs. 27 into Eq. 22, the kinetic energy contribution to E xc,x can be obtained as follows: T,X T s,x = E xc,x e 2 E xc,x e 2 2,m 28a = m E xc,x m 2,e 2 28b 2 = E xc,x 2. 28c e 2,m The right-hand sides of Eqs. 28 contain the derivatives of E xc,x that are calculable enough. Thus, the right-hand side of Eq. 22 is rewritten by three kinds of calculable expressions Eqs. 28. Equations 28 show the exact relations between E xc,x and T,X T s,x. If we get one of the two functionals, then each of Eqs. 28 can be utilized as a constraint to another functional. Also, focusing only on the right-hand sides of Eqs. 28a 28c, these equalities can be utilized as constraints in developing the approximate form of E xc,x. It should be noted that the kinetic energy contribution to E xc of the conventional DFT is given in terms of the derivative with respect to not e 2, m, or 2 but the Bohr radius alone 58. The above-mentioned relations that include E xc,x alone can be obtained only by expressing the kinetic energy contribution in terms of multiple kinds of derivatives. The difference T,X T s,x is alternatively estimated by using the virial relation 14. The virial relation in the ECS theory has been derived under the assumption that X 0 r is transformed into d X 0 r by the coordinate scaling of electrons, where is the scale factor. Under this assumption, the virial relation for isolated systems is given by 14 T 0,X 0 T s,x + E xc 0,X rr E xc,x r = 0 X=X 0dr + X 0 r r d +3 E xc,x =0. Xr = 0 X=X 0dr 29 Combining Eqs. 28 with Eq. 29, we can eliminate T,X T s,x and obtain the exact relations that include only E xc,x, PHYSICAL REVIEW A 79, r = 0 X=X 0dr X 0 r r d +3 E xc,x Xr = 0 X=X 0dr E xc 0,X 0 0 rr E xc,x = E xc 0,X 0 e 2 E xc 0,X 0 e 2 2,m 30a = m E xc 0,X 0 m 2,e 2 30b 2 = E xc 0,X c e 2,m Equations 30 can also be utilized as constraints in developing the approximate form of E xc,x. In addition, we may employ Eqs. 30 as sum rules in the so-called virial method 75,76, in which the virial relation 29 is used in calculating the orbital-dependent exchange-correlation potential. It is preferable to use Eqs. 30 in place of Eq. 29, because we do not need T,X T s,x in Eqs. 30. Next, we shall divide Eqs. 28 into two parts, i.e., exchange and correlation parts, because the exchange and correlation energy functionals, E x,x and E c,x, are often developed in parts as in the case of the GGA and VEA Similarly to the DFT, E x,x of the ECS theory is defined by E x,x =,XŴ,X U. 31 Since it is the minimizing Slater determinant in the definition of T s,x,,x satisfies the equation Min,X Tˆ=,XTˆ,X. By applying the Lagrange multiplier method, the equation for,x can be obtained as follows: Tˆ + v s rˆrdr + a s r Xˆ rdr,x = s,x, 32 where v s r, a s r, and s are the Lagrange multipliers. In the same way as the discussion below Eq. 23, we can show that both v s,xr and a s,xr are independent of e 2, m, and 2. Therefore, it is concluded that,x is also independent of them. Differentiating Eq. 31 with respect to e 2, m, or 2 with the aid of the above-mentioned fact, we get 2 E x,x e E x,x e 2 =0, 33a 2,m E x,x m e 2, 2 =0, 33b E x,x 2 =0. 33c e 2,m

6 KATSUHIKO HIGUCHI AND MASAHIKO HIGUCHI Concerning the correlation energy functional, it is given by E c,x=e xc,x E x,x. By using Eqs. 33, Eqs. 28 can therefore be rewritten as T,X T s,x = E c,x e 2 E c,x e 2 2,m 34a = m E c,x m 2,e 2 34b 2 = E cx 2. 34c e 2,m These exact relations Eqs. 33 and 34 can be employed as constraints in devising the approximate form of E x,x and E c,x as well as in the virial method 75,76. IV. DISCUSSION In this section, we shall discuss the assumption that Xˆ r is independent of e 2, m, and 2. Let us consider the case in which xˆr is chosen as the basic variable instead of Xˆ r. Here recall that xˆr is the operator of the usual physical quantity and is related to Xˆ r as Xˆ r=xˆr/ fe 2,m, 2. The universal functional of this case is defined by f,x =Min,x Tˆ +Ŵ. The search area in this definition is the set of antisymmetric wave functions that yield both r and xr. It should be noticed that antisymmetric wave functions in this set simultaneously yield both r and Xr. Therefore, the set of antisymmetric wave functions yielding both r and xr coincide with that of antisymmetric wave functions yielding both r and Xr. Consequently, it can be concluded that Min,x Tˆ +Ŵ=Min,X Tˆ +Ŵ. This equation leads to,x =,X, 35 where,x is the minimizing wave function in the definition of f,x. The same discussion holds for the minimizing Slater determinant in the kinetic energy functional. Namely, we have,x =,X, 36 where,x is the minimizing Slater determinant in Min,x Tˆ =t s,x. Using Eqs. 35 and 36, we can conclude the following equations: xc,x = E xc,x, x,x = E x,x, c,x = E c,x, where xc,x is the exchange-correlation energy functional of the ECS theory including xr as a basic variable. PHYSICAL REVIEW A 79, On the other hand, by means of the relation xr = fe 2,m, 2 Xr, the functional of r and Xr is formally obtained from that of r and xr. For example, we have xc,x = xc, fe 2,m, 2 X xc,x, 40 where xc,x is the functional of r and Xr. From Eqs. 37 and 40, we can immediately construct xc,x =E xc,x and evaluate it by means of exact relations that are derived in the previous section. These discussions give the conditions that indicate which type of quantities can be included in the formula of the kinetic energy contribution. For example, if we choose the paramagnetic current density as Xr, then the present results are applicable as mentioned above. However, that is not the case if the sum of the paramagnetic current density and spincurrent density is chosen as Xr. Thus, the assumption introduced is not a restriction in usual cases. V. CONCLUDING REMARKS In the framework of the ECS theory, we have derived the generalized version of Bauer s relation 59. The relation represents the difference between expectation values of an arbitrary quantity with respect to the ground-state wave function and that with respect to the Kohn-Sham determinant of the ECS theory. This expression is similar to that of the conventional DFT, except that the arbitrary physical quantity Xr is added to it as the basic variable. The generalized Bauer s relation is successfully used to derive the kinetic energy contribution to the exchangecorrelation energy functional. Since the exchange-correlation energy functional of the ECS theory can be expressed as a function of the Bohr radius and e 2, three kinds of expressions for the kinetic energy contribution are derived. That is, the kinetic energy contribution is expressed in terms of three kinds of derivatives with respect to e 2, m, or 2, while it is in terms of the derivative with respect to the Bohr radius alone in Bass s expression of the conventional DFT 58. Here, we shall give a comment on the importance of this type of work. As mentioned in Sec. I, exact relations have been effectively used in developing the approximate form of the exchange-correlation energy functional in the DFT, SDFT, CDFT, and so on. For example, the GGA is also developed with the aid of many kinds of exact relations. It is one of the established methods to utilize as constraints exact relations that are fulfilled by the exchange-correlation energy functional. Many works concerning exact relations have been done so far. We derive exact relations in this paper. Their exact relations will be utilized as constraints in developing and/or evaluating the approximate form of the exchange-correlation energy functional of the ECS theory. Furthermore, by means of these expressions, we can eliminate the kinetic energy contribution term in the virial relation of the ECS theory. The resultant relations Eqs. 30 include not the kinetic energy contribution but the exchangecorrelation energy functional alone, which also could be strong sum rules not only for the ECS theory but also for the virial method 75,76. Let us go into details. In

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8 KATSUHIKO HIGUCHI AND MASAHIKO HIGUCHI 64 K. Capelle and G. Vignale, Phys. Rev. Lett. 86, K. Capelle and G. Vignale, Phys. Rev. B 65, K. Capelle and V. L. Libero, Int. J. Quantum Chem. 105, W. Kohn, A. Savin, and C. A. Ullrich, Int. J. Quantum Chem. 100, C. A. Ullrich, Phys. Rev. B 72, P. W. Ayers and W. T. Yang, J. Chem. Phys. 124, T. Heaton-Burgess, P. Ayers, and W. T. Yang, Phys. Rev. Lett. 98, H. Eschrig and W. E. Pickett, Solid State Commun. 118, PHYSICAL REVIEW A 79, S. Pittalis, S. Kurth, N. Helbig, and E. K. U. Gross, Phys. Rev. A 74, A different type of expression has been derived by Nagy 54. Whereas in the expression by Nagy denotes both the scale factor of electron coordinates and the coupling constant of electron-electron interaction, in Eq. 22 stands for a scalar field that is coupled with Ô. 74 R. M. Dreizler and E. K. U. Gross, Density Functional Theory Springer-Verlag, Berlin, M. Kodera, K. Higuchi, A. Narita, and M. Higuchi, J. Phys. Soc. Jpn. 76, M. Kodera, K. Higuchi, A. Narita, and M. Higuchi, Phys. Rev. A 78,